| Advanced computational techniques used to quantify the numerical accuracy are essential components for robust and reliable use of these simulations in engineering. Over the last two decades, a number of endeavors have focused on the challenge for development and improvement of error estimation techniques and adaptive discretization methods. A particular contribution in recent years has been dedicated to the development of an a-posteriori finite element error estimation technique termed the 'bound method '. This novel technique provides fast yet reliable, accurate, and efficient bounds to the quantity of interest, termed the 'output' here-in, expressed as a functional of the field solutions for the underlying partial differential equations (PDEs). Furthermore, the method naturally yields a local error indicator, which estimates the numerical error for given numerical models, leading to the construction of 'ideal' meshes for computations. The bound method in this work is applied to the multi-physical, multi-scale and multi-dimensional microfluidic phenomena, in particular, the electro-osmotic flow in microchannels. An appropriate mathematical representation of the electro-osmotic flow in microchannels can be expressed by multi-physical PDEs which span inter-disciplinary physical phenomena such as the fluid mechanics, heat/mass transfer, surface phenomena as well as electro-mechanics. The primarily goal of this thesis is the development of the novel computational tools applied to multi-disciplinary applications governed by the non-linear and non-coercive PDEs such as steady incompressible Navier-Stokes and Energy equations. The second goal of this thesis is the improvement of the bound method enhanced by advanced numerical strategies such as an adaptive refinement, the direct equilibration, and parallel computing. The final and ultimate goal of this thesis is the application of the advanced bound method in the context of the design and optimization for the microfluidics in a microchannel network to better analyze and predict microfluidic flow motion and species transport phenomena. Three numerical models, namely the non-slip velocity model, the slip velocity model, and the EO-velocity model, for simulating the electro-osmotic flow in microchannels are considered in this work. The bound method based on the above three numerical models for the electro-osmotic flow in microchannels will be constructed. The bound method will be applied for the electro-osmotic flow in various microchannels. Furthermore, its performance will be illustrated in terms of numerical accuracy and computational cost. |
